= Solution
The <Generalized Stokes theorem> applied to the <closed differential one-form> gives
$$
\boxed{\rho_1(k)+\rho_2(\bar ak)+\rho_3(ak)=0.}
$$
Equivalently, the <global relation> is
$$
\sum_{j=1}^3E(-ic_jk)\Phi_j(c_jk)
=-\frac i2\sum_{j=1}^3E(-ic_jk)\Psi_j(c_jk).
$$
The identical outward <Neumann boundary data> make the three known transforms equal:
$$
\Psi_j(k)=\Psi(k)=\int_{-l/2}^{l/2}e^{(k+\lambda/k)s}f(s)ds.
$$
To reduce the unknown traces to a single function, use rotation invariance and uniqueness. For $\lambda>0$ the difference $u$ of two solutions has zero outward derivative and
$$
\int_D|\nabla u|^2dA+4\lambda\int_D|u|^2dA=0,
$$
by <Green's first identity>. Thus $u=0$. Rotation by $a$ preserves the equation, the domain and the oriented boundary data, so it preserves $q$; hence $q^{(1)}=q^{(2)}=q^{(3)}=q$ and $\Phi_j=\Phi$. The <global relation> becomes the \b[single equation with three spectral unknowns]
$$
E(-ik)\Phi(k)+E(-i\bar ak)\Phi(\bar ak)+E(-iak)\Phi(ak)
=-\frac i2[E(-ik)\Psi(k)+E(-i\bar ak)\Psi(\bar ak)+E(-iak)\Psi(ak)].
$$
The three unknowns are evaluations of the same boundary transform, not three freely specified boundary data.
The printed allowance of all real $\lambda$ requires a qualification. At $\lambda=0$, existence requires $3\int_{-l/2}^{l/2}f(s)ds=0$ by the <divergence theorem>, and solutions differ by a constant. Fixing a rotation-invariant normalization gives uniqueness and the same rotational conclusion. In fact the unnormalized solution is already invariant: its rotation difference is constant, and three successive rotations make three times that constant zero. For $\lambda<0$, uniqueness fails precisely when $-4\lambda$ is an <eigenvalue> of the <Neumann Laplacian>. At these parameters identical data do not force an arbitrary solution to be rotationally invariant. Averaging any solution over the three rotations produces a solution with the common trace used here; the difference is a homogeneous <Neumann eigenfunction>, or a sum of them. Part (e) describes both solvability and this remaining freedom rather than assuming uniqueness for every real parameter.
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